Understanding the definition of current I can solve almost all my textbook problems related to current electricity using formulas but it seems in the end, I don't understand what current is. The bookish definition of current is:

Current is defined to be the amount of charge passed through a cross sectional area of a conductor per unit time.

I want to know, if how I think of current is right and, if possible, I would very much appreciate, if a better way of thinking current is provided.
Let us take a conductor and fix a cross sectional area and suppose there are $n$ free electrons on one side of the cross section which will eventually hit the cross section. We take a stop watch in our hand. And now when the electrons start moving, we start the stop watch. Now the electrons will randomly hit the cross section one by one or a few at once. After $1$ second, we stop the timer. We see that $x$ electrons have gone penetrating that cross section. Since charge of each electron is $e$,then $xe$ charge has flown in that one second. So current will be $xe$ A.
Is my line of thought correct? Specially do the points marked in italics matter? or just somehow the number of electrons penetrating that cross section in one second matter?
 A: Yes this is a correct picture of current. Related concepts here could be drift velocity, which you might find helpful to supplement your understanding.
A: 
Now the electrons will randomly hit the cross section one by one or
a few at once

No way one by one, or a few at once, is sufficient to obtain a measurable current.
Since the charge on a single electron in coulombs (C) is about 1.602 x 10$^{-19}$ C and since 1 A of current equals 1 coulomb of charge crossing the section per second, that means you need to have 6.24 x 10$^{18}$ electrons crossing the section per second for one ampere of current.

Since charge of each electron is $e$,then $xe$ charge has flown in
that one second. So current will be $xe$ A.

Provided it is understood that the charge on each electron is 1.602 x 10$^{-19}$ C, then yes the current will be $xe$.
Hope this helps.
A: I prefer the definition of current that is best applied to maxwells equations.
$I=\iint \vec{J} \cdot \vec{da}$
$\vec{J} = \rho \vec{V}$ (Current density)
$\rho$ is the charge density at a point in space ($\frac{Q}{m^3}$)
$\vec{V}$ is the velocity of the charge density at a point in space ($\frac{m}{s}$)
The flux transport integral is a measure of how much the flow of the quantity $\vec{J}$, flows directly through a surface. In this case, the cross section of a wire.
This integral has units
$\frac{Q}{m^3} \cdot  \frac{m}{s} \cdot m^2$
$= \frac{Q}{s}$
A: 
Is my line of thought correct?

For the most part yes, it's correct, but I think you are not realizing that you are calculating the average current over a period of one second.

or just somehow the number of electrons penetrating that cross section in one second matter?

In the situation you are describing, it doesn't matter how (one by one or in a group) or at which point of this one-second period the electrons are crossing the surface. If the flow at a constant rate, or if all $x$ cross at once the last microsecond, the result is still $xe$ A. This, again, is because you're taking the average number of electrons crossing the surface oin one second. If you want to know the instant current over that second, you need to know how many electrons are crossing the surface at each point in time. Note that the current is no longer a single value, but a function of time, since the number of electrons crossing the surface will be a function of time:
$$I(t) = en_e(t)$$
where $n_e(t)$ is the number of electrons crossing the surface at a point in time. Using this expression requires you to know exactly how many electrons are crossing the surface at any given time. In other words, it does matter if electrons are crossing one by one or a lot at a time.
For electric circuits however, it is consider that the flow of electrons $n_e(t)$ is almost constant, which means that current itself is constant. Because of this, if you take the average current over a period of time (as you did in your example) it will be the same as the the instant current at any time, but note that this is only the case BECAUSE we are considering that the electrons cross the surface as a constant rate.
